[過去ﾛｸﾞ] 現代数学の系譜 工学物理雑談 古典ガロア理論も読む49 (658ﾚｽ)
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59: 2017/12/30(土)12:02 ID:T5iI1wtu(1) AAS
スレ主は今年も進歩ゼロでしたとさ
60: 2017/12/30(土)13:17 ID:dSbKeTYf(1) AAS
ここまで酷いとホントにゴミだよ
なんで数学板にいるの？ってレベル
61: 2017/12/31(日)11:42 ID:yDllqZzl(1/2) AAS
>>58
これってスレ主？
62: 2017/12/31(日)12:15 ID:65THZoXS(1) AAS
ズレ主を表す今年の漢字2017は？
「誤」「乱」「偽」「劣」「愚」
63: 2017/12/31(日)15:12 ID:yDllqZzl(2/2) AAS
スレ主の反応が無い。
冬休み、うつ、飽きた
どれだ？
64: 2018/01/01(月)01:44 ID:9ORABeV3(1/2) AAS
スレ主がこのまま消えますように
65: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:04 ID:dCRrvhl7(1/27) AAS
皆さま、どうも。スレ主です。（＾＾
明けまして、おめでとうございます。
新年も、よろしくお願いします。ｍ（＿ ＿）ｍ
66: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:06 ID:dCRrvhl7(2/27) AAS
>>53
ID:gcYWyS10 さん、沢山レスありがとう
貴方のレスは、レベル高いね
あとで、じっくり読むよ（＾＾
67(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:07 ID:dCRrvhl7(3/27) AAS
で、勝手ながら、年末年始に読んだ関連を貼るよ（＾＾
まず、関連参考：検索でヒットしたので貼る。
BaireCategory.pdfの”3. Pointwise limits of continuous functions.”に、「422に書いた定理」の関連記述
「Theorem. If f : R → R is a pointwise limit of continuous functions,
then Df is Fσ meager (that is, a countable union of closed sets with empty interior).
(In particular, by Baire's theorem, f is continuous on a dense subset of R.)」とあり（当たり前か？ （＾＾ ）
外部ﾘﾝｸ[html]:www.math.utk.edu
MATH 447- Advanced Calculus I- Fall 2016- A. FREIRE
(or: ANALYSIS IN R^n)
(抜粋)
外部ﾘﾝｸ[pdf]:www.math.utk.edu
Sets of discontinuity and Baire's theorem Baire Category Notes (5 problems) (the problems are HW8, due Friday 11/4)A. FREIRE 2016
(抜粋)
1. Sets of discontinuity. For f : R → R, we define
Df = {x ∈ R; f is not continuous at xg:
3. Pointwise limits of continuous functions.
Theorem. If f : R → R is a pointwise limit of continuous functions,
then Df is Fσ meager (that is, a countable union of closed sets with empty interior).
(In particular, by Baire's theorem, f is continuous on a dense subset of R.)
Proof. We know Df = ∪ n>=1 D1/n (see Section 1), so it suffices to show
that the closed sets Dε have empty interior, for any ε > 0.
By contradiction, suppose Dε contains an open interval I.
We'll find an open interval J ⊂ I disjoint from Dε!
Let fn → f pointwise on R, with each fn : R → R continuous.
For each N >= 1, consider the set:
CN = {x ∈ I; (∀m, n >= N)|fm(x) - fn(x)| <= ε/3}.
Clearly ∪ N>=1 CN = I (by pointwise convergence). QED
(引用終り)
つづく
68(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:08 ID:dCRrvhl7(4/27) AAS
>>67 つづき
（上記の関連参考：出典URL）
外部ﾘﾝｸ[html]:www.math.utk.edu
Math 447 Fall 2016- A. FREIRE
TOPICS
PART I: Topology
Supplementary handouts (for advanced students):
(adapted from more advanced classes and not yet in final form)
外部ﾘﾝｸ[pdf]:www.math.utk.edu
Definitions and Theorems from General Topology
外部ﾘﾝｸ[pdf]:www.math.utk.edu
Locally compact Banach spaces are finite dimensional (includes 4 problems)
外部ﾘﾝｸ[pdf]:www.math.utk.edu
Spaces of Continuous Functions (outdated)
外部ﾘﾝｸ[pdf]:www.math.utk.edu
Stone-Weierstrass theorem-notes (includes 6 problems)
外部ﾘﾝｸ[pdf]:www.math.utk.edu
Ascoli-Arzela-Notes (final-included 7 exercises with solutions, and 11 extra problems.)
外部ﾘﾝｸ:www.math.utk.edu
Alex Freire
Department of Mathematics
University of Tennessee
（終り）
つづく
69(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:10 ID:dCRrvhl7(5/27) AAS
>>68 つづき
あと、いま、「422に書いた定理」に、似た文献を見つけて読んでいる。（＾＾
”I-DENSITY CONTINUOUS FUNCTIONS Krzysztof Ciesielski他 1994”
これ、出版されていて、アマゾンでもヒットした
疑問が二つ
１）Proposition 1.1.1. の「Given ε > 0 there is a δ > 0 such that {x ∈ (x0 − δ, x0 + δ) : |f(x) − f(x0)| ≧ ε} ∈ J.」で、普通のεδ論法だと、 |f(x) − f(x0)| ＜ ε と不等号の向きが逆になると思うが、誤植か？ σ-ideal を考えているから、これで良いのか？ どうも良いみたいだが
２）Corollary 1.1.6. の「(ii): There exists a residual set K such that f|K is continuous.*2」で、f|Kは、Theorem 1.1.4.の”(ii): There exists a set K ∈ J such that the restricted function f|Kc is continuous.”の記載ぶりとの比較から、f|Kcの誤記かなと思ったり？ 意味が全く違ってくる
これにも、「422に書いた定理」の関連記述あり（後述）
つづく
70(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:10 ID:dCRrvhl7(6/27) AAS
>>69 つづき
外部ﾘﾝｸ[pdf]:www.math.wvu.edu
I-DENSITY CONTINUOUS FUNCTIONS Krzysztof Ciesielski他 1994- 被引用数: 84
(抜粋)
CHAPTER 1
The Ordinary Density Topology
1.1. A Simple Category Topology
To gain some insight into what is happening with limits like this, it is useful
to generalize this idea to a topological setting.
A nonempty family J ⊂P(X) of subsets of X is an ideal on X if A ⊂ B and
B ∈ J imply that A ∈ J and if A∪B ∈ J provided A,B ∈ J. An ideal J on X
is said to be a σ-ideal on X if ∪n∈N An ∈ J for every family {An : n ∈ N} ⊂ J.
Let J be an ideal on R and Ｔo be the ordinary topology on R. The set
T (J) = {G ＼ J : G ∈ Ｔo, J ∈ J}
is a topology on R which is finer than Ｔo. The following proposition is evident
from the definitions.
Proposition 1.1.1. Let J be a σ-ideal on R and T (J ) be as above. For
f : (R, T (J )) → (R, Ｔo) and x0 ∈ R the following statements are equivalent to
each other.
(i): f is continuous at x0.
(ii): Given ε > 0 there is a δ > 0 such that
{x ∈ (x0 − δ, x0 + δ) : |f(x) − f(x0)| ≧ ε} ∈ J.
つづく
71(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:11 ID:dCRrvhl7(7/27) AAS
>>70 つづき
Theorem 1.1.4. Let J be a σ-ideal and f : R → R. The following statements
are equivalent.
(i): The function f is J -continuous J -a.e.
(ii): There exists a set K ∈ J such that the restricted function f|Kc is continuous.
Furthermore, if the ideal J contains no interval, then the following statement is
equivalent to (i) and (ii)
(iii): There exists a function g : R → R such that f = g J -a.e. and g is continuous in the ordinary sense J -a.e.
Proof. The fact that (ii) implies (i) is obvious. Suppose (i) is true and let
f be J -continuous on a set M = Jc for J ∈ J. For each n ∈ N and x ∈ M,
by Proposition 1.1.1(ii) there is an open interval I(n, x) and a J(n, x) ∈ J such
that
x ∈ I(n, x) \ J(n, x) ⊂ f−1((f(x) − 1/n, f(x) + 1/n)).
For each fixed n, there must be a countable sequence xn,m ∈ M such that
M ⊂∪m∈N I(n, xn,m).
Let
K = J ∪ ∪ n,m∈N J(n, xn,m) ∈ J.
If x ∈ Kc and ε > 0, then there must exist natural numbers n and m such that
2/n < ε and x ∈ I(n, xn,m). Then |f(x) − f(xn,m)| < 1/n so that
f(x) ∈ (f(xn,m) − 1/n, f(xn,m) + 1/n) ⊂ (f(x) − ε, f(x) + ε)
つづく
72(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:11 ID:dCRrvhl7(8/27) AAS
>>71 つづき
and
I(n, xn,m) ∩ Kc ⊂ f−1((f(x) − ε, f(x) + ε)).
Hence, f|Kc is continuous at x.
To prove the last part of the theorem, note first that (iii) implies (ii) even
without the restriction that J contains no interval. Now suppose that J contains
no interval and that f,K are as in (ii). Define
(1) G(x) = lim sup t→x,t∈Kc f(t)
and
(2) g(x) = G(x) when G(x) is finite,
or = f(x) otherwise.
In particular, it follows from (ii) that f|Kc = g|Kc . Let x ∈ Kc and ε > 0.
According to (ii) there is a δ > 0 such that
(3) |g(y) − g(x)| = |f(y) − f(x)| < ε/2
whenever y ∈ (x − δ, x + δ) ∩Kc. If z ∈ (x − δ, x + δ) ∩K, then the assumption
that K can contain no nonempty open set implies the existence of a sequence
{zn : n ∈ N} ⊂ (x − δ, x + δ) ∩ Kc
such that f(zn) → G(z). Hence, by (3), G(z) is finite, so g(z) = G(z) and
|g(z) − g(x)| ? ε/2 < ε. Therefore, g is continuous at x. QED
つづく
73(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:12 ID:dCRrvhl7(9/27) AAS
>>72 つづき
The following example is interesting in light of the previous theorem.
Example 1.1.5. Let I be the σ-ideal consisting of all first category subsets of
R. I-continuity is often called qualitative continuity [26]. It is well-known in
this case that f is a Baire function if, and only if, f is qualitatively continuous I-a.e.
In particular, combining Example 1.1.5 with Theorem 1.1.4 yields the following
well-known corollary, which will be useful in the sequel.
Corollary 1.1.6. Let f : R → R. The following statements are equivalent.
(i): f is a Baire function.
(ii): There exists a residual set K such that f|K is continuous.*2
(iii): f is qualitatively continuous I-a.e.
In the case of Lebesgue measure, the following is true.
*2 A set is residual if its complement is first category. This is often called comeager.
つづく
74(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:12 ID:dCRrvhl7(10/27) AAS
>>73 つづき
If condition (i) in Theorem 1.1.4 is strengthened to everywhere, the following corollary results.
Corollary 1.1.8. Let J be a σ-ideal which contains no nonempty open set.
A function f : R → R is continuous everywhere if, and only if, it is J -continuous everywhere.
Proof. If f is continuous, then it is clearly J -continuous. So, suppose f is
J -continuous everywhere, x0 ∈ R and ε > 0. Using Proposition 1.1.1(ii), there
must be an ordinary open neighborhood G0 of x0 such that
F0 = {x ∈ G0 : |f(x) − f(x0)| > ε} ∈ J.
Suppose there is an x1 ∈ F0. Choose δ > 0 such that
δ < |f(x1) − f(x0)| − ε.
As before, there exists an ordinary open neighborhood G1 ⊂ G0 of x1 such that
F1 = {x ∈ G1 : |f(x1) − f(x)| > δ} ∈ J.
It is clear that G1 ⊂ F0 ∪ F1 ∈ J, because |f(x1) − f(x0)| > ε + δ. But, this
implies J contains a nonempty open set, which contradicts the condition placed
on J in the statement of the corollary. This contradiction shows that F0 = Φ.
The preceding corollary demonstrates that global J -continuity may not be a
very useful concept. In particular, it is worthwhile noting for future reference
that global I-continuity and global N-continuity are no different than ordinary continuity.
(引用終り)
つづく
75: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:13 ID:dCRrvhl7(11/27) AAS
sage
76(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:13 ID:dCRrvhl7(12/27) AAS
>>74 つづき
（上記の関連参考：出典URL）
外部ﾘﾝｸ:www.math.wvu.edu
Krzysztof Chris Ciesielski, Ph.D. Professor of Mathematics at Department of Mathematics, West Virginia University and Adjunct Professor at Medical Image Processing Group, Dept. of Radiology, Univ. of Pennsylvania.
(抜粋)
Books:
(with L. Larson and K. Ostaszewski) I-density continuous functions, Memoirs of the AMS vol. 107 no 515, 1994; MR 94f:54035.
(引用終り)
外部ﾘﾝｸ:www.amazon.co.jp
I-Density Continuous Functions (Memoirs of the American Mathematical Society) (英語) Krzysztof Ciesielski (著),? Lee Larson (著),? Krzysztof Ostaszewski (著) 1994/1/1
外部ﾘﾝｸ:www.jstor.org
JOURNAL ARTICLE I-density Continuous Functions Krzysztof Ciesielski, Lee Larson and Krzysztof Ostaszewski Real Analysis Exchange Vol. 15, No. 1 (1989-90), pp. 13-15 Published by: Michigan State University Press
（終り）
つづく
77(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:14 ID:dCRrvhl7(13/27) AAS
>>76 つづき
（参考：用語解説）
外部ﾘﾝｸ:en.wikipedia.org
Ideal (set theory)
(抜粋)
In the mathematical field of set theory, an ideal is a collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal (this codifies the idea that an ideal is a notion of smallness), and the union of any two elements of the ideal must also be in the ideal.
More formally, given a set X, an ideal I on X is a nonempty subset of the powerset of X, such that:
1. Φ ∈ I
2.if A∈ I and B⊆ A, then B∈ I, and
3.if A,B∈ I, then A ∪ B∈ I
Some authors add a third condition that X itself is not in I; ideals with this extra property are called proper ideals.
Ideals in the set-theoretic sense are exactly ideals in the order-theoretic sense, where the relevant order is set inclusion. Also, they are exactly ideals in the ring-theoretic sense on the Boolean ring formed by the powerset of the underlying set.
Contents
1 Terminology
2 Examples of ideals
2.1 General examples
2.2 Ideals on the natural numbers
2.3 Ideals on the real numbers
2.4 Ideals on other sets
3 Operations on ideals
4 Relationships among ideals
5 See also
6 References
(引用終り)
つづく
78(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:14 ID:dCRrvhl7(14/27) AAS
>>77 つづき
外部ﾘﾝｸ:en.wikipedia.org
σ-ideal Sigma-ideal (Redirected from Σ-ideal)
(抜粋)
In mathematics, particularly measure theory, a σ-ideal of a sigma-algebra (σ, read "sigma," means countable in this context) is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent application is perhaps in probability theory.
Let (X,Σ) be a measurable space (meaning Σ is a σ-algebra of subsets of X). A subset N of Σ is a σ-ideal if the following properties are satisfied:
(i) O ∈ N;
(ii) When A ∈ N and B ∈ Σ , B ⊆ A ⇒ B ∈ N;
(iii) {A_n}_{n∈N }⊆ N→ ∪ _{n∈N }A_n∈ N.
Briefly, a sigma-ideal must contain the empty set and contain subsets and countable unions of its elements. The concept of σ-ideal is dual to that of a countably complete (σ-) filter.
If a measure μ is given on (X,Σ), the set of μ-negligible sets (S ∈ Σ such that μ(S) = 0) is a σ-ideal.
The notion can be generalized to preorders (P,?,0) with a bottom element 0 as follows: I is a σ-ideal of P just when
(i') 0 ∈ I,
(ii') x ? y & y ∈ I ⇒ x ∈ I, and
(iii') given a family xn ∈ I (n ∈ N), there is y ∈ I such that xn ? y for each n
Thus I contains the bottom element, is downward closed, and is closed under countable suprema (which must exist). It is natural in this context to ask that P itself have countable suprema.
A σ-ideal of a set X is a σ-ideal of the power set of X. That is, when no σ-algebra is specified, then one simply takes the full power set of the underlying set. For example, the meager subsets of a topological space are those in the σ-ideal generated by the collection of closed subsets with empty interior.
(引用終り)
つづく
79(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:15 ID:dCRrvhl7(15/27) AAS
>>78 つづき
外部ﾘﾝｸ:en.wikipedia.org
Ideal
(抜粋)
Mathematics
Ideal (ring theory), special subsets of a ring considered in abstract algebra
Ideal, special subsets of a semigroup
Ideal (order theory), special kind of lower sets of an order
Ideal (set theory), a collection of sets regarded as "small" or "negligible"
Ideal (Lie algebra), a particular subset in a Lie algebra
Ideal point, a boundary point in hyperbolic geometry
Ideal triangle, a triangle in hyperbolic geometry whose vertices are ideal points
(引用終り)
つづく
80(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:16 ID:dCRrvhl7(16/27) AAS
>>79 つづき
外部ﾘﾝｸ[pdf]:www.artsci.kyushu-u.ac.jp
典型的連続関数のDini微分（著者最終稿）斎藤新悟 実解析学シンポジウム2009報告集，pp. 25-33.
（上記の関連参考：出典URL）
外部ﾘﾝｸ[html]:www.artsci.kyushu-u.ac.jp
斎藤新悟 出版物
外部ﾘﾝｸ:www.artsci.kyushu-u.ac.jp
斎藤新悟 九州大学基幹教育院准教授 1981年大阪府生まれ 東京大学理学部数学科卒業
つづく
81(2): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:16 ID:dCRrvhl7(17/27) AAS
>>80 つづき
（以前のスレから関連抜粋）
スレ46 2chｽﾚ:math
＜引用＞
外部ﾘﾝｸ[pdf]:www.unirioja.es
DIFFERENTIABILITY OF A PATHOLOGICAL FUNCTION, DIOPHANTINE APPROXIMATION, AND A REFORMULATION OF THE THUE-SIEGEL-ROTH THEOREM JUAN LUIS VARONA 2009
This paper has been published in Gazette of the Australian Mathematical Society, Volume 36, Number 5, November 2009, pp. 353{361.
(抜粋)
So, in this paper we
are going to analyze the dierentiability of the real function
fν(x) =0 if x ∈ R ＼ Q,
or =1/q^ν if x = p/q ∈ Q, irreducible,
for various values of ν ∈ R.
Theorem 1. For ν > 2, the function fν is discontinuous (and consequently
not dierentiable) at the rationals, and continuous at the irrationals. With
respect the dierentiability, we have:
(a) For every irrational number x with bounded elements in its continued fraction expansion, fν is differentiable at x.
(b) There exist infinitely many irrational numbers x such that fν is not differentiable at x.
Moreover, the sets of numbers that fulfill (a) and (b) are both of them uncountable.
Theorem 2. For ν > 2, let us denote
Cν = { x ∈ R : fν is continuous at x },
Dν = { x ∈ R : fν is dierentiable at x }.
Then, the Lebesgue measure of the sets R ＼ Cν and R ＼ Dν is 0, but the four sets Cν, R ＼ Cν, Dν, and R ＼ Dν are dense in R.
(引用終り)
つづく
82(2): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)17:17 ID:dCRrvhl7(18/27) AA×
>>81
83(1): 2018/01/01(月)17:38 ID:WRx3yiBV(1) AAS
>>82
>R中のQのように稠密分散で、
>R＼Qは、”a nonempty open set”の集まりになるけれども
？
84(1): 2018/01/01(月)17:48 ID:HicRQN2S(1/2) AAS
おっちゃんです。
今日は午前4時に散歩したら、新聞配達のお姉ちゃんが自転車で配達していた。
今は意識もうろうとしていて、もうお寝んねタイム。
85(1): 2018/01/01(月)18:10 ID:HicRQN2S(2/2) AAS
まあ、深夜に散歩するのも案外日常とは違う面白い光景が見られる。
深夜にコンビニに行く人も時々見かける。
昼間の車の排気ガスで汚れた空気とは違い、昼間程汚れていない新鮮な空気は吸えるな。
それじゃ、おっちゃん寝る。
86(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)19:50 ID:dCRrvhl7(19/27) AAS
Thomae（「ポップコーン」）関数の絵が面白いので、ご紹介。
外部ﾘﾝｸ:arxiv.org
外部ﾘﾝｸ:arxiv.org
Number-theoretic aspects of 1D localization: "popcorn function" with Lifshitz tails and its continuous approximation by the Dedekind eta S. Nechaev, K. Polovnikov (Submitted on 22 Feb 2017 (v1), last revised 26 Feb 2017 (this version, v2))
(抜粋)
We discuss the number-theoretic properties of distributions appearing in physical systems when an observable is a quotient of two independent exponentially weighted integers.
The spectral density of ensemble of linear polymer chains distributed with the law ?fL (0<f<1),
where L is the chain length, serves as a particular example.
At f→1, the spectral density can be expressed through the discontinuous at all rational points, Thomae ("popcorn") function.
We suggest a continuous approximation of the popcorn function, based on the Dedekind η-function near the real axis.
Moreover, we provide simple arguments, based on the "Euclid orchard" construction, that demonstrate the presence of Lifshitz tails, typical for the 1D Anderson localization, at the spectral edges.
We emphasize that the ultrametric structure of the spectral density is ultimately connected with number-theoretic relations on asymptotic modular functions.
We also pay attention to connection of the Dedekind η-function near the real axis to invariant measures of some continued fractions studied by Borwein and Borwein in 1993.
(引用終り)
87(1): 2018/01/01(月)20:25 ID:9ORABeV3(2/2) AAS
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88: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/01(月)20:35 ID:dCRrvhl7(20/27) AAS
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Figure 5: Plots of everywhere continuous f1(x) = -ln |η(x + iε)| (blue) and discrete f2(x) = Π/(12ε) g^2(x) (red) for ε = 10^-6 at rational points in 0 < x < 1.
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